Theorem 0nnq 11048

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5692	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 11036	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4010	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3909	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 200	1 ⊢ ¬ ∅ ∈ Q

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147  ∀wral 3080  ∅c0 4281   class class class wbr 5108   × cxp 5656  ‘cfv 6566  2^nd c2nd 8034  Ncnpi 10968   <_N clti 10971   ~_Q ceq 10975  Qcnq 10976

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174  df-xp 5664  df-nq 11036

This theorem is used by:  adderpq  11080  mulerpq  11081  addassnq  11082  mulassnq  11083  distrnq  11085  recmulnq  11088  recclnq  11090  ltanq  11095  ltmnq  11096  ltexnq  11099  nsmallnq  11101  ltbtwnnq  11102  ltrnq  11103  prlem934  11157  ltaddpr  11158  ltexprlem2  11161  ltexprlem3  11162  ltexprlem4  11163  ltexprlem6  11165  ltexprlem7  11166  prlem936  11171  reclem2pr  11172